For example, when cloth, yarn, paper and the like are colored to a desired color or to the same shade as that of a given sample, it is necessary to prepare coloring compositions including a plurality of colorants which would be able to reproduce the given color. However, if the shade of the cloth actually colored by such coloring compositions is not the same as the given color, the coloration results in failure, and a lot of cloth thus colored would not be able to fulfil the requirements for user.
In the actual operation of coloring factories, therefore, a well-skilled engineer chooses a desired combination of colorants among a great number of known colorants and estimates the shade of a material or substrate colored by a mixture of the chosen colorants having a suitable ratio. Then, from various kinds of colorant compositions thus estimated, he selects the most suitable composition or recipes and actually colors a test piece by the selected colorant composition. The colored test piece is compared to the given reference colored sample. If the color of the test piece is not matched with the colored reference sample, the engineer attemps to modify the recipe of the colorant composition and repeats the same procedure as that mentioned above until the strict color matching is obtained.
The above color matching operation is very important and indispensable prior to actual coloring operation in coloring factories. As seen from the above, however, the manual color matching operation is very troublesome, and the precision and speed thereof greately depend upon the experience and ability of the coloring engineer. In addition, although the engineer is extremely well-skilled, his treatment ability is very small because he has to pass through many trial-and-errors until he succeeds to reproduce the shade of the given sample. In practice, the engineer can treat only a few cases of color matching per day. Therefore, the production capacity of the coloring factories is completely and extremely limited because of the number and the treatment power of the color matching engineers.
In order to reduce the trial-and-errors for color matching, therefore, methods have previously been proposed to estimate, by use of a computer, the shade realized by a proposed recipe of colorants without actual coloring. If there are determined a few kinds of colorant recipes which would enable the shade of the given sample to be reproduced or closely approached before the actual coloring, the times of the trial-and-errors for color matching can be decreased to a large extent, so that the amount of color matching treated by one engineer would be greately increased.
At present, thereore, great efforts are paid to develop methods for computer aided color matching and to study the optical property of colored layers. On the other hand, spectrophotometers and microcomputers, which are required for computer aided color matching, have been recently greatly advanced and widely spread.
Thus, computer aid color matching is used to formulate the colorant composition in many coloring factories.
This computer aided color matching is generally called "CCM" in this specification. This CCM performed at present is foundamentally based on the theory of colored layer proposed by Kubelka and Munk in 1931.
In the Kubelka-Munk theory, a colored layer is analyzed by using one-dimension model in the direction of thickness and by assuming that each colorant contained in the colored layers has an inherent absorption coefficient K and scattering coefficient S. As a result, reflectivity of colored layer of different thicknesses is expressed in terms of the absorption coefficient K and the scattering coefficient S.
Assuming that the reflected light does not contain the light reflected by the base material or substrate to which a colorant is applied and also assuming that the reflected light is not influenced by the boundary between the air and the surface of the colored layer, the following Kubelka-Munk equation is concluded between the absorption coefficient K, the scattering coefficient S and the reflectivity R.sub..infin. of the colored layer at any given wavelength as follows: EQU K/S=(1-R.sub..infin.).sup.2 /2R.sub..infin. ( 1)
When the CCM is actually performed, the above equation is used in combination with the following Duncan equation: ##EQU1## where (K/S).sub.mix is the ratio of K to S of a layer colored by a recipes containing n colorants which have the absorption coefficient K.sub.i and the scattering coefficient S.sub.i, respectively and which are mixed with each other at the concentration C.sub.i respectively. Ko and So are respectively the absorption coefficient and the scattering coefficient of a base material or substrate bearing the colorants, such as vehicle, fiber and the like.
In fiber dyeing, since the scattering coefficient S.sub.i of a colorant is negligibly small as compared with the scattering coefficient So of the fiber itself, it is assumed that S.sub.i .ident.O and the Duncan equation (2) is modified as follows: ##EQU2##
This modified equation is also widely utilized in the paint and plastics industries, because it is applicable to the case of coloring a vehicle, resin or the like which includes a large amount of opaque white pigment such as titanium dioxide.
In the CCM operation, the (K/S).sub.mix is calculated by using the above equation (3) and the calculated (K/S).sub.mix is applied to the following equation which can be derived from the equation (1). EQU (K/S).sub.mix =(1-R.sub.mix).sup.2 /2R.sub.mix ( 4)
As a result, there can be obtained the reflectivity R.sub.mix of the layer colored by the mixture of n colorants having the absorption coefficient Ki and a negligible scattering coefficient, respectively and also having concentration C.sub.i, respectively.
Thereafter, the reflectivity R.sub.mix is compared with the reflectivity R.sub.s of the given color sample at all wavelengths in visible spectrum. If R.sub.mix is not consistent with R.sub.s, the aforementioned calculation and comparison procedure are repeatedly performed by changing the concentration C.sub.i of the respective colorants, so that the reflectivity R.sub.mix of the resulting colorant mixture is consistent with the reflectivity R.sub.s of the given color sample at all wavelengths in visible spectrum. This is called "Isomeric Matching Method".
In addition to the Isomeric Matching Method, there is known a so-called "Metameric Matching Method". In this method, calculation is made to obtain, from the reflectivities R.sub.mix and R.sub.s of the proposed colorant recipe and the color sample at different wavelengths, the tristimulus values X.sub.mix, Y.sub.mix, Z.sub.mix and X.sub.s, Y.sub.s, Z.sub.s of Commission Internationale de l'Eclairage (CIE) under a certain CIE standard illuminant, and then there is calculated the concentrations C.sub.i of individual colorants making it possible to coincide the two sets of tristimulus values with each other.
The CCM has been actually performed by using the above methods. Consequently, the color matching operation is greatly labor-saved and time-saved in the coloring industries.
However, the previous methods actually involve various problems.
Namely, if it is desired to obtain a recipe of colorants for a deep color, the conventional CCM (which uses, for example, the aforementioned Kubelka-Munk equation, or other equations such as the Atkins equation, the Pineo equation, the Fink-Jensen equation, the Love-Oglesby-Gailey equation, etc. as shown in "COLOR research and application" Vol. 2, No. 3, 1977,) cannot provide the recipe which reproduces the same shade as that of a given color sample. Here, the term "deep color" means that the color has at least one strong absorption band in the visible spectrum. If a desired color becomes deeper, the inconsistency between the actual color and the estimated color becomes large.
The reason for this is considered as follows:
In a weak absorption band, the linear relation can be obtained between the colorant concentration and the value (K/S) calculated by using the Kubelka-Munk equation. However, in a strong absorption band, a linear relation cannot be obtained between the colorant concentration and the calculated value (K/S). FIGS. 1a, 1b and 2a, 2b show such two examples. In these Figures, the weak absorption band is 600 nm in wavelength and the strong absorption band is 440 nm. In addition, the solid line shows the relation between the concentration and the value (K/S) calculated in accordance with the conventional methods, and the dotted line shows the regression line of the solid line shown in the same Figures.
The above mentioned inconsistency is considered to result from the fact that the value (K/S) derived from the Kubelk-Munk equation or the other conventional method does not a good linearity against the concentration of colorant in the strong absorption region.
For the above reason, the CCM utilized at present cannot be successfully applied to coloring of high deepness.
Furthermore, when the color of a given concentration is estimated by using the spectral reflectivity of a reference colored material manufactured on the basis of a predetermined colorant recipe, the estimated spectral reflectivity is not correct because of lack of linearity as mentioned above. Therefore, strength evaluation is very troublesome in the quality control of colorants. Namely, it has to prepare a lot of colored smaples differing in concentration, and then to examine the relation between the concentration and the shade of the actually colored test piece.